Geometry Flashcards
vertex, vertices
sommet
acute angle
< 90°
obtuse angle
> 90°
area of a triangle
A = (1/2)bh
b : base
h : height (altitude)
altitude of a triangle
line that goes through the vertex and is perpendicular to the opposite side
legs
sides of a triangle that meet at the right angle
in a right triangle, the legs represent two of its altitudes. If one leg is the base, the other leg is the altitude, and the area equals 1/2 times the product of the legs
median of a triangle
goes from a vertex to the midpoint of the opposite side
Pythagorean theorem
hypotenuse² = leg² + leg²
Pythagorean triplets
sets of integers that satisfy the theorem 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
we could also multiply any of these fundamental triplets by any nb to create a new set of three nb 6, 8, 10 9, 12, 15 12, 16, 20 15, 20, 25 18, 20, 25 ...
scale factor
Similar triangles
scale factor = ratio: lengths of sides are proportional so k is the factor by which all lengths in the small figure were multiplied to arrive at the lengths in the large figure
when lenghts are multiplied by scale factor k, area is multiplied by k²
similar figures
same shape but different size angles are equal two triangles are similar if they simply share two angles sides are proportional scale factor
Isosceles right triangle
angles of 45°-45°-90°
sides of 1-1-√2
two equal legs, and (hypotenuse) = (√2)*(leg)
equilateral triangle + altitude = right triangles
angles of 30°-60°-90°
sides of 1-√3-2
hypotenuse = 2 * short leg
long leg = √3 * short leg
rhombuses
parallelograms with four sides equal and perpendicular diagonals
area of a trapezoid
find the average of the bases, and multiply this by the height
A = ((b1 + b2)/2)h
or subdivide the trapezoid into a central rectangle and two side right triangles (in a symmetrical trapezoid, those two side right triangles will be congruent)
area of a quadrilateral
A = bh
pentagon
a polygon with 5 vertices, sides, and diagonals
can be broken into three triangles
sum of the angles = 540°
hexagon
a polygon with 6 vertices and sides
9diagonals
can be broken into four triangles
sum of the angles = 720°
regular polygon
equilateral and equiangular: most symmetrical and well-balanced example of a category
regular triangle = equilateral
regular quadrilateral = square
regular pentagon: each angle = 108°
regular hexagon: each angle = 120°
regular octagon: each angle = 135°; sum of angles = 1080°
chord
a segment with 2 endpoints on the circle
the longest chord passes through the center and is called a diameter
circumference of a circle
c = πd c = 2πr
π approximations
3.14
22/7
area of a circle
A = πr²
inscribed angle and central angle relationship with the arc they intercept (circle properties)
the measure of a central angle is the measure of the arc it intercepts
the measure of an inscribed angle is HALF the measure of the arc it intercepts
Circle propertiesr
if two sides of a triangle are radii, the triangle is isosceles
a central angle has the same measure as the arc it intercepts
equal length chords intercept equal arcs
an inscribed angle has half the measure of the arc it intercepts
an angle inscribed in a semicircle is 90°
two inscribed angles intersecting the same chord on the same side are equal
a tangent line is perpendicular to a radius at the point of tangency
proportion of an arclength
arclength/2πr(circumference) = angle/360
ratio of areas to the ratio of angles
area of sector/πr² = angle/360
total volume and surface area of a cube
total volume and surface area of a rectangle
V = s^3; SA = 6s² V = hwd; SA = 2hw + 2hd + 2wd
3D version of the Pythagorean theorem
AD² = AB² + BC² + CD²
diagonal of a square
space diagonal of a cube
side*√2
side*√3
volume of a cylinder
V = πr²h
total surface area of a cylinder
2πr²+2πrh
2x surface of circle + surface of rectangle
A cylinder neatly encloses a sphere, so that the curve of the cylinder touches the sphere in a circle at its widest, and the top & bottom faces of the cylinder are tangent to the sphere. If the volume of the sphere is V = 4/3 * πr^3, what fraction of the cylinder does the sphere occupy?
the radius of the sphere and that of the cylinder must be the same, r
the height of the cylinder must be as tall as the sphere, so it must equal the diameter, h = 2r
cylinder; V = πr²h = πr²2r = 2πr^3
if we divide 4/3 * πr^3 by 2πr^3, the πr^3 parts will cancel, and we will just be left with (4/3)/2 = 2/3
Between two similar figures, if the length increases by scale factor k, the area increases by… and the volume increases by…
ratio of the lengths: k
ratio of the surfaces: k²
ratio of the volumes: k^3
A bathroom floor is a rectangle, 2m by 3m. it is to be tiled with square tiles, 4cm on a side. How many tiles will it take to fill the floor? (1m = 100cm)
area = 6m²(100cm/1m)² = 60 000cm^3
divide this by 4cm² (= divide by 4 twice)
60000/4 = 15000
15000/4 = (16000 - 1000) /4 = 3750
A swimming pool has the capacity of a rectangular solid 5m x 8m on the surface and 2m deep. The pool is filled with grains of sand. If each grain of sand is 1mm^3, and 1m = 1000mm, how many grains of sand are in the pool?
V = 2*5*8 = 80m^3 (80m^3)(1000mm/1m)^3 = 80(10^3)^3 = 80(10^9) = 8(10^10)
